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\begin{document}

\section{Navier-Stokes incompressible}

\subsection{Formulation forte}

\begin{eqnarray}
\rho_{f} \frac{\partial u_{f}}{\partial t}_{|_{ALE}} + \rho_{f} ( u_{f} - w ) \nabla u_{f} - \nabla \cdot \sigma_{f} &=& 0 \\
\nabla \cdot u_{f} &=& 0
\end{eqnarray}

with $\sigma_{f} =  -p Id + 2\mu D_{f}$ et $D_{f}= \frac{1}{2}( \nabla u_{f} + (\nabla u_{f})^{T})$



\subsection{Formulation variationnelle}

$(u_{f},p)$ trial ; $(v_{f},q)$ tests functions :


\begin{eqnarray}
  \int_{\Omega(t)} \rho_{f} \frac{\partial u_{f}}{\partial t}_{|_{ALE}} \cdot v_{f}
  + \int_{\Omega(t)} \rho_{f} ( u_{f} - w ) \nabla u_{f} \cdot v_{f} 
  - \int_{\Omega(t)} (\nabla \cdot \sigma_{f}) \cdot v_{f}  &=& 0 \\
(\nabla \cdot u_{f} ) q &=& 0
\end{eqnarray}


Integration par parties : 

\begin{eqnarray}
  \int_{\Omega(t)} \rho_{f} \frac{\partial u_{f}}{\partial t}_{|_{ALE}} \cdot v_{f}
  + \int_{\Omega(t)} \rho_{f} ( u_{f} - w ) \nabla u_{f} \cdot v_{f} 
  + \int_{\Omega(t)} \sigma_{f} : \nabla v_{f}
  - \int_{\partial \Omega} \sigma_{f} \vec{n} \cdot v_{f}  &=& 0 \\
(\nabla \cdot u_{f} ) q &=& 0
\end{eqnarray}


On peut aussi écrire :
\begin{eqnarray}
\int_{\Omega(t)} \sigma_{f} : \nabla v_{f} = \int_{\Omega(t)}-p Id : \nabla v_{f} + \int_{\Omega(t)}2\mu D_{f} = \int_{\Omega(t)}-p \nabla \cdot  v_{f} + \int_{\Omega(t)} 2\mu D_{f}
\end{eqnarray}

et : 
\begin{eqnarray}
\int_{\Omega(t)}  \rho_{f} \frac{\partial u_{f}}{\partial t}_{|_{ALE}} \cdot v_{f}
= \frac{d}{dt}\int_{\Omega(t)}\rho_{f} u_{f} \cdot v_{f} - \int_{\Omega(t)}\rho_{f} (\nabla \cdot w) \cdot v_{f}
\end{eqnarray}

\paragraph{}
Application à l'étude 2d : \\
$\Gamma_{E}$ : entrée \\
$\Gamma_{I}$ : interface fluide-structure \\
$\Gamma_{S}$ : sortie

\paragraph{}
Les conditions aux limites sont :
\begin{itemize}
\item $\sigma_{f}=0$ sur $\Gamma_{S}$ (contrainte libre)
\item $u_{f} = w$ sur $\Gamma_{I}$  (vitesse du maillage fluide)
\item $\sigma_{f} = g$ sur $\Gamma_{E}$ (onde de pression)
\end{itemize}


\begin{eqnarray}
  \int_{\Omega(t)} \rho_{f} \frac{\partial u_{f}}{\partial t}_{|_{ALE}} \cdot v_{f}
  + \int_{\Omega(t)} \rho_{f} ( u_{f} - w ) \nabla u_{f} \cdot v_{f} 
  + \int_{\Omega(t)} \sigma_{f} : \nabla v_{f} \\
  - \int_{\Gamma_{E}} g \cdot v_{f}
  - \int_{\Gamma_{I}} \sigma_{f} \vec{n} \cdot v_{f}
  + \int_{\Gamma_{I}} \frac{\gamma}{ h_{face}} (u_{f}-w) \cdot v_{f}  &=& 0 \\
(\nabla \cdot u_{f} ) q &=& 0
\end{eqnarray}



\end{document}